Fowler Long Scale Calculator

This a neat pocket-watch-like mechanical slide rule calculator, with
an effective scale length of about 750mm.

Theory

Multiplication

The theory of operation of the Fowler is similar to a conventional slide
rule (for which see here) but the moves are
different in practice. The Fowler has only one logarithmic scale so you
cannot add the distances by sliding one scale against another. Instead you
use a pair of pointers to store a distance read from the scale, representing
one argument, and then use the scale again to add the second argument, and
the scale again to read off the result.

The diagrams below show the principle, but using a 1-dimensional linear
scale, rather like using a conventional slide rule with the slide missing
and a pair of dividers to carry distances.

The lower part is the base or fixed part. It carries a fixed pointer F and
a second pointer A which can be moved relative to the base. The top part,
the scale, also moves relative to the base, independent of A.

To multiply 2 by 3, first move the scale so that 2 is over the fixed
pointer.

Then move pointer A until it aligns with the index on the scale. The
distance from A to F now represents the scale length (logarithm) for 2.

Next move the scale again to bring the second argument 3 against pointer
A. Then the scale length for 3 is added to the scale length for 2, stored
as the distance A to F, and the answer 6 can be read off the scale opposite
pointer F.

On the real calculator the scale is actually a circle or a group of
circles. Pointer A and the scale can both be rotated relative to the base
part. Between them pointers A and F store a distance along the scale.

Division

To divide 6 by 3, first move the scale to set 6 under F.

Then move pointer A to align with 3.

Then move the slide again to bring the index mark to align with A and
read the result 2 under F.

Chaining

Note that both multiplication and division leave the result under the
F pointer, and that they start with the multiplicand or dividend under F, so
chaining in any sequence is simple.

Examples

Now let's look at the real calculator, for example we'll try to multiply
2 by 3, initially using the short scale. First turn the top knob to align 2
on the outer scale with the fixed pointer F at the top.

Then use the side knob to bring the moving pointer A up to the index or
"10" mark on the scale.

Finally turn the top knob again to bring 3 on the outer scale under the
moving pointer A. The result, 6, can be read on the outer scale under the
fixed pointer F at the top.

We can then repeat the operation using the long scale. The operations are
just the same except we have to choose which of the six turns of the long
scale we have to use. First bring up the 2 under the fixed pointer. 2,
for which we use the mark labelled "20" (remembering that slide rules do
not give powers of ten, you have to work them out for yourself), is found on
the second circle out from the centre.

Then as before move the pointer A to the index of the scales. This is
always the same place, for both the short and long scales.

Lastly bring the "30" mark under the pointer A. The "30" mark is on the
third scale out from the centre. The result is then any of the numbers on
the six long scales under fixed pointer F: about 13, 19, 28, 41, 60 or 88.

From the short scale calculation we know the result is about 60, so we
read the result, which we know is 6, from the 5th scale.

Although it is tedious to have to calculate everything twice, it is
good practice as it is easy to make mistakes on this sort of mechanical
calculator, and the self-checking from doing the same calculation in
slightly different ways is useful.

Other functions

There are interesting interactions between the back and front scales. The
back has only one pointer, fixed at the top, but the scales rotate on the
side knob, at the same time as the pointer A on the front. On the
back are another basic short scale and based on that: reciprocal, logarithms,
square roots and trig functions. These can be used as simple look-ups.
But the linking with the front allows other things to work. For example
putting 2 under F on the front and 2 under F at the back makes 4 appear
under A on the front. Setting index under F at the front makes A at the
front agree with F at the back. More about these when I have a better
understanding.

Accuracy and other considerations

One point I don't understand is why the long scale is provided as six
separate concentric circles rather than a 6-turn spiral. There seems no good
reason to do this, unless it was to get round a patent, and it makes reading
and setting awkward near a value where the jump occurs from one scale to
the next. In the picture below, coming towards the end of the first circle
is the numbered mark for 1.4, followed by tick marks for 1.41 to 1.46. The
tick marks for 1.47 to 1.49 and the numbered mark for 1.5 are on the second
scale. The transition point is actually 1.468, the sixth-root of 10. Not
only is this confusing to read, but interpolating between 1.46 and 1.47 is
difficult.

It's fairly easy to interpolate the scales to three figures, with a partial
fourth figure in the lower half of the scale. The mechanical accuaracy is
good and the cursor lines and scale markings are narrow and easy to align.
It should be possible to achieve around 0.2% per calculation, based on
0.5mm total error for the four steps.